Showing papers on "Operator algebra published in 1971"
TL;DR: In this article, the critical indices for the two-dimensional Ising model were derived from an assumed form of an operator algebra which describes how the product of two fluctuating variables may be reduced to a linear combination of the basic variables.
Abstract: A previous publication showed how the critical indices for the two-dimensional Ising model could be derived from an assumed form of an operator algebra which describes how the product of two fluctuating variables may be reduced to a linear combination of the basic fluctuating variables. In this paper, the previously used algebra is derived from the Onsager solution of the two-dimensional Ising model. The calculation makes use of a "disorder" variable which is mathematically the result of applying the Kramers-Wannier transformation to the Ising-model spin variable. The average of products of spin and disorder variables are evaluated at the critical point for the special case in which all the variables lie on a single straight line. The ordering of these variables on the line determines a "quantum number" $\ensuremath{\Gamma}$ such that the average is nonzero only for $\ensuremath{\Gamma}=0$. Composition rules for this quantum number are derived and used to develop an algebra for the multiplication of complex variables at the critical point. Arguments are given to suggest the identifications of elements of the algebra as the spin, the energy density, the Kaufman spinors, and a stress density. The result of this calculation is the operator algebra which formed the starting point of the previous paper.
421 citations
125 citations
01 Feb 1971
TL;DR: In this article, it was shown that the norm of a hermitian element in a complex Banach algebra is equal to the spectral radius of the element, which is a generalisation of Bernstein's theorem which gives a bound on the derivative of an entire function along the real line.
Abstract: We prove that the norm of a hermitian element in a Banach algebra is equal to the spectral radius of the element. An element h in a complex Banach algebra with identity (of norm 1) is said to be hermitian if Jjexp iahjj = 1 for all real a [61, [3, Definition 5.1]. I. Vidav uses a Phragmen-Lindel6f theorem to show that the numerical radius [3, Definition 2.1] of a hermitian element is equal to its spectral radius [6, p. 123, Hilfssatz 3], [3, Theorem 5.10]. We show that the norm of h +f1 is equal to the spectral radius of h+f1 for h a hermitian element and ,B a complex number (Proposition 2). The proof uses a generalisation of Bernstein's theorem which gives a bound on the derivative of an entire function along the real line. F. F. Bonsall and M. J. Crabb [2 ] have recently given an elementary proof of our Proposition 2 when ,B is zero (which is equivalent to ,B real). In Lemma 5 and Proposition 6 we construct a norm on the algebra of polynomials, in one indeterminate x, which is maximal with respect to the property that x is hermitian of norm one. An entire function F is said to be of order R if log log M(c) R = lim sup aoo log a where M(a) denotes sup { F(z) zIz
63 citations
47 citations
09 Feb 1971
TL;DR: In this article, a condition under which a linear operator belongs to G sub p is determined, i.e., if p = or > 1 it is a Banach algebra under a suitable norm.
Abstract: : Let H be a Hilbert space and let G sub p be the set of all linear operators A on H such that the trace of (A star)(A sup p/2) is finite. G sub p is a two-sided ideal in the algebra of all bounded operators on H; if p = or > 1 it is a Banach algebra under a suitable norm. The principal aim of this paper is to determine a condition under which a linear operator belongs to G sub p. (Author)
42 citations
TL;DR: In this article, the set of all projections and unitaries in a von Neumann algebra factor A were studied from the homotopical point of view relative to the operator norm topology.
Abstract: The set of all projections and the set of all unitaries in a von Neumann algebra factorA are studied from the homotopical point of view relative to the operator norm topology.
38 citations
TL;DR: In this article, the Von Neumann algebra model for quantum systems has been applied to the analysis of partially ordered vector spaces, and some properties of these properties are derived and applications to quantum theory are discussed.
Abstract: Recent work of Davies and Lewis has shown how partially ordered vector spaces provide a setting in which the operational approach to statistical physical systems may be studied. In this paper, certain physically relevant classes of operations are identified in the abstract framework, some of their properties are derived and applications to the Von Neumann algebra model for quantum theory are discussed.
34 citations
TL;DR: In this article, the phase operators ''cosine'' and ''sine'' are characterized as a special and peculiar class of tridiagonal operators, defined on an abstract separable Hilbert space.
Abstract: The phase operators ``cosine'' and ``sine'' are characterized as a special and peculiar class of tridiagonal operators, defined on an abstract separable Hilbert space. The two disjoint sets of these operators, which lie on the unit sphere of the algebra of bounded operators, are convex and closed in the uniform topology. The whole treatment gives a new and systematic aspect to the quantum mechanical oscillator phase problem.
30 citations
TL;DR: In this article, it was shown that every operator has a non-trivial invariant subspace, and every operator other than a multiple of the identity has a hyper-invariant hyper-subspace.
Abstract: It is well-known, and easily verified, that each of the following assertions implies the preceding ones. (i) Every operator has a non-trivial invariant subspace. (ii) Every commutative operator algebra has a non-trivial invariant subspace, (iii) Every operator other than a multiple of the identity has a non-trivial hyperinvariant subspace. (iv) The only transitive operator algebra on is Note. Operator means bounded linear operator on a complex Hilbert space , operator algebra means weakly closed algebra of operators containing the identity, subspace means closed linear manifold, a non-trivial subspace is a subspace other than {0} and , a. hyperinvariant subspace for A is a subspace invariant under every operator which commutes with A, a transitive operator algebra is one without any non-trivial invariant subspaces and denotes the algebra of all operators on .
23 citations
TL;DR: In this paper, it was shown that if the real field admits a certain type of observable (together with other structure which seems to be physically justified), then the real fields contain the real subfield as a subfield.
Abstract: Corresponding to any irreducible proposition system L in general quantum mechanics there is a division ring D with an anti‐automorphism * and a vector space (V, D) over D with a definite sesquilinear form φ such that L is isomorphic to the set of φ closed subspaces of (V, D). The main task remaining in connecting the general quantum mechanics to the conventional quantum theory in a complex Hilbert space is to give physical arguments which force D to be the complex field. In this paper it is shown that if L admits a certain type of observable (together with other structure which seems to be physically justified), then D contains the real field as a subfield. Steps are then indicated that can be taken to move from the reals to the complexes or quaternions.
23 citations
TL;DR: The perturbation theory of Hille and Phillips for semigroups of bounded linear operators on a Banach space is modified to apply to the semigroup of positive traceclass operators encountered in quantum statistical mechanics as discussed by the authors.
Abstract: The perturbation theory of Hille and Phillips for semigroups of bounded linear operators on a Banach space is modified to apply to the semigroups of positive traceclass operators encountered in quantum statistical mechanics.
TL;DR: In this paper, a theory for perturbations of Hermitian operators with respect to completely continuous operators is presented, where singular and characteristic numbers are derived for operators from the von Neumann algebra and their minimax properties are derived.
Abstract: By means of the concept of Segal measure, defined on projectors (and thus on subspaces associated with a von Neumann algebra) we introduce the concept of relative compactness of sets and, on this basis, the concept of operators completely continuous with respect to the von Neumann algebra and the Segal measure The article is concerned with the formal structure of the theory of this class of operators: the general theorem of Calkin is obtained on the uniqueness of the ideal with respect to completely continuous operators; a theory is constructed for perturbations of Hermitian operators with respect to completely continuous ones; singular and characteristic numbers are introduced for operators from the von Neumann algebra and their minimax properties are derived; some characterizations are introduced in terms of completely continuous operatorsBibliography: 10 items
TL;DR: The notion of hermitian functions on complex Banach algebras was introduced in this paper, where it was shown that every functional f in the dual A' of A can be decomposed as f = h + ik, where h and k are hermitians.
Abstract: The hermitian functionals on a unital complex Banach algebra are defined here to be those in the real span of the normalized states (tangent functionals to the unit ball at the identity). It is shown that every functional f in the dual A' of A can be decomposed asf= h + ik, where h and k are hermitian functionals. Moreover, this decomposition is unique for every f E A' iff A admits an involution making it a C*-algebra, and then the hermitian functionals reduce to the usual real or symmetric functionals. A second characterization of C*-algebras is given in terms of the separation properties of the hermitian elements of A (real numerical range) as functionals on A'. The possibility of analogous theorems for vector states and matrix element functionals on operator algebras is discussed, and potential applications to the representation theory of locally compact groups are illustrated. Introduction. As the abstract indicates, we introduce here the notion of a hermitian functional on an arbitrary complex Banach algebra (B-algebra) with norm-one unit; these functionals generalize real measures on a compact Hausdorff space (qua functionals on C(X)) and the real or symmetric functionals on a unital C*-algebra. Our definition exploits the earlier involution-independent geometrical characterization of the positive functionals or states on a C*-algebra, due to Bohnenblust and Karlin [2], as well as Lumer's strikingly successful use of this to define states on general B-algebras and hermitian members of such algebras [7]. We show rather easily here that the dual space A' of A is the complex span of the states (a mild improvement of the Bohnenblust-Karlin vertex theorem [2]) and thus that every f E A' can be expressed as f= h + ik for suitable " real "and "imaginary" hermitian parts h and k. The entire dual space of A is thus brought into play in the geometrical study of A, making possible two considerably deeper duality characterizations of C*-algebras: A admits a C* involution iff A' decomposes as a real direct sum A' = H(A') + iH(A') of hermitians, or equivalently iff the hermitians in A (or their complex span) separate points in A'. (This brings the study Received by the editors March 20, 1970 and, in revised form, December 15, 1970. AMS 1969 subject classifications. Primary 4650, 4660; Secondary 4665, 2260.
TL;DR: In this article, the theory of nonlinear semi-groups is used to study the Thirring and Federbush models in quantum field theory and the existence and uniqueness of the global solution is proved.
Abstract: Some field equations, suggested by the Thirring and Federbush models in quantum field theory are studied in a two-dimensional space-time. The theory of nonlinear semi-groups is used. The unknowns are functions whose values are bounded operators on a Hilbert space. The existence and uniqueness of the global solution is proved.
TL;DR: In this paper, various applications of the theory of hyperdifferential operators to quantum mechanics are discussed, and a concise summary of the relevant aspects of this theory is presented, and then used to derive a variety of operator identities, expansions, and solutions to differential equations.
Abstract: In this paper, various applications of the theory of hyperdifferential operators to quantum mechanics are discussed. A concise summary of the relevant aspects of the theory is presented, and then used to derive a variety of operator identities, expansions, and solutions to differential equations.
TL;DR: Theorem 0.1 as discussed by the authors gives an affirmative solution, in a large number of cases, to the following problem: When does there exist a strongly continuous unitary representation a -* U(a) of G on X' such that each U (a) induces p(a)?
Abstract: This paper gives an affirmative solution, in a large number of cases, to the following problem. Let M be a von Neumann algebra on the Hilbert space X, let G be a topological group, and let a -p(a) be a homomorphism of G into the group of *-automorphisms of M. Does there exist a strongly continuous unitary representation a -U(a) of G on X such that each U(a) induces p(a)? 0. Introduction. The following problem arises in quantum mechanics. Let R be a von Neumann algebra on the Hilbert space X 9 a topological group, and a -* p(a) a representation of G into the group of *-automorphisms of M. When does there exist a strongly continuous unitary representation a -* U(a) of G on X' such that each U(a) induces (p(a)? Roughly speaking, this question asks: Does a Hamiltonian exist? The main result of this paper is to prove the following theorem, which gives an affirmative answer to the above question under fairly general hypotheses. THEOREM 0.1. Let R be a semifinite von Neumann algebra on the separable Hilbert space X Suppose that the commutant of 9 has no finite portion. Let G be a locally compact group whose topology is second countable, and let a -* p(a) be a representation of G as a group of center-fixing *-automorphisms of St. Suppose that a is continuous for all T in S and x, y in X Then there exists a strongly continuous unitary representation a -U(a) of G on A' such that (p(a)(T) U(a)TU(a-1) (a in G, T in St). One cannot omit the hypothesis that the commutant of the properly infinite portion of 9 is also properly infinite, for there exists a II,, factor , with 9' finite, which possesses nonspatial *-automorphisms (see Kadison [6]). I do not know if an analogue of Theorem 0.1 is true in case R has a portion of type III ?1 contains a proof of Theorem 0.1 in case 9 is a factor. ?2 and ?3 are devoted to preliminary topics in direct integral theory and Borel structure theory of various objects. The proof of Theorem 0.1 is given in ?4 by combining the results of the previous three sections. Received by the editors April 6, 1970 and, in revised form, July 27, 1970. AMS 1968 subject classifications. Primary 4600; Secondary 4665.
TL;DR: In this article, the authors developed the mathematical machinery necessary in order to describe systematically the commutation and anticommutation relations of the field algebras of an algebraic quantum field theory of the fermion type.
Abstract: We develop the mathematical machinery necessary in order to describe systematically the commutation and anticommutation relations of the field algebras of an algebraic quantum field theory of the fermion type. In this context it is possible to construct a skew tensor product of two von Neumann algebras and completely describe its type in terms of the types of the constituent algebras. Mathematically the paper is a study of involutory automorphisms of If*-algebras, of particular importance to quantum field theory being the outer involutory automorphisms of the type III factors. It is shown that each of the hyperfinite type III factors studied by Powers has at least two outer involutory automorphisms not conjugate under the group of all automorphisms of the factor.
TL;DR: In this article, the axioms of measurements introduced by Ludwig are formulated and studied in the framework of operator algebras, and it is shown that a concrete C * algebra with identity satisfies the axiom of sensitivity increase of effects if and only if it is a von Neumann algebra.
Abstract: The axioms of measurements introduced by Ludwig are formulated and studied in the framework of operator algebras. It is shown that a concrete C*‐algebra with identity satisfies the axiom of sensitivity increase of effects if and only if it is a von Neumann algebra; although a von Neumann algebra satisfies the axiom of decompossability of ensembles, however, the axiom of components of the mixtures of two ensembles is true only if a von Neumann algebra is a factor of type In (n < + ∞). It is also verified that the set of decision effects, which is proved to be a subset of projections of a von Neumann algebra, has similar lattice structure of quantum mechanics, and its connection with quantum logic in the sense of Varadarjan is also figured out.
TL;DR: In this article, a representation of symmetry transformations motivated by the functional formulation of quantum field theory is rigorously discussed in a functional Hilbert space, where the set of generating functionals is equipped with an inner product by means of the Friedrichs-Shapiro-integral.
Abstract: Abstract A representation of symmetry transformations motivated by the functional formulation of quantum field theory is rigorously discussed in a functional Hilbert space. The set of generating functionals is equipped with an inner product by means of the Friedrichs-Shapiro-integral and completed to an Hilbert space. Unitarity, continuity, and reducibility are investigated for the symmetry operations in this space. Also non-unitary transformations are considered.
01 Jun 1971
TL;DR: In this article, it was shown that if a linear algebra with identity has two equivalent unital algebra norms, |.| 1 and |.• 2, whose corresponding numerical radii, v 1 and v 2, are equal on the whole algebra, are not necessarily equal.
Abstract: This paper owes its origin to the following question posed by A. M. Sinclair, “If a linear algebra with identity has two equivalent unital algebra norms, |.| 1 and |.| 2 , whose corresponding numerical radii, v 1 and v 2 , are equal on the whole algebra, are |.| 1 and |.| 2 related? Are they, for example, necessarily equal?” We do not give a complete answer to this question but are able to give sufficient conditions on algebras of operators for v 1 = v 2 to imply |.| 1 = |.| 2 That this implication does not hold for an arbitrary algebra with identity is demonstrated by means of a counter-example. The result for operator algebras is used to deduce some essentially non numerical range results for equivalent operator norms.
TL;DR: It is shown that for any C(*)-algebra A of operators on a separable Hilbert space, there is a sequence of self-adjoint operators such that {x(n)} is monotone decreasing and strongly convergent to x.
Abstract: It is shown that for any C*-algebra A of operators on a separable Hilbert space, there is, for each self-adjoint operator x in the strong closure of A, a sequence {xn} of self-adjoint operators, each of which is the strong limit of a monotone increasing sequence of self-adjoint operators from A, such that {xn} is monotone decreasing and strongly convergent to x.